\Chapter{Experimental Setup and Methods}
\label{chap:experimental}
\ifpdf
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    							{Structure/cpgsFigs/}}
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\section{Numerical Simulation}
\label{sec:numerical}

\subsection{FDTD}
\label{sec:FDTD}
Although analytical solutions to the responses of some plasmonic structures can be formulated, as demonstrated in \fref{sec:analytic}, the majority of nano-scale structures are too complex to make analytical solutions over the structure possible. However, a theoretical model of such structures is nevertheless desirable, in order to verify observed experimental results, to produce new designs for structures, and to refine design parameters of structures to suit particular spectral characteristics or other design constraints. Several techniques for modelling such micro- and nano-scale structures exist which may be considered semi-analytical, in that they apply numerical methods to solve analytic expressions for the electric fields. Such techniques are used to predict scattering profiles and spectra of nanoplasmonic structures, as well as the modes of metal waveguides and other objects\cite{Zia2005,Berini2000,Berini2001}. One of the most commonly used is the \FDTD\ method, the basic algorithm of which dates back to a 1966 paper by Kane Yee \cite{Yee1966}. The \FDTD\ algorithm is conceptually simple to understand, and is also simple to implement in software. The method is used extensively, and at a wide range of length and frequency scales, including applications from radio and microwave antenna design \cite{Zhao2007,Toland1993}, to simulations of nano-scale structures including photonic crystals and plasmonic antennas.

The basic algorithm is to discretize a simulation volume and the \EM\ fields it contains, using finite-difference approximations to the partial derivatives in the Maxwell equations. The method is still macroscopic in the sense that it uses the continuous-media approximations of using permittivity and permeability values for materials' response to \EM\ fields. Electric field vectors are defined at the points of a cartesian grid, while magnetic field vector elements are defined at points interleaved between the electric field components, as shown in \fref{fig:FdtdVoxel}. The algorithm steps forward in a leapfrog manner, with the electric field vectors $\E$ at time $t=n\Delta t$ being used to calculate the magnetic field vectors $\X{H}$ at time $t=\fn{n+\frac{1}{2}}\Delta t$.

To illustrate the algorithm, consider again the Maxwell curl equations, \ref{eq:MaxwellCurlE} and \ref{eq:MaxwellCurlH}. Considering a region with no sources of current, we can set $\rho_{\rm{ext}}=0$. Rearranging, we may obtain the equations in the following forms:

\begin{eqnarray}
\label{eq:FDTDH}\partiald{\X{H}}{t}&=&-\mu_0\Curl{\E}\\
\label{eq:FDTDE}\partiald{\E}{t}&=&\frac{1}{\epsilon}\fn{\frac{1}{\epsilon_0}\Curl{\X{H}}-\E\partiald{\epsilon}{t}}
%\label{eq:FDTDE}\partiald{\E}{t}&=&\frac{1}{\epsilon_0\epsilon}\fn{\Curl{\X{H}}-\sigma\E}
\end{eqnarray}

differentials of a function $f\fn{\X{r},t}$ at position $\X{r}=\fn{i\Delta x,j\Delta y,k\Delta z,}$ and at a time $t=n\Delta t$ are approximated by the finite difference method as follows

\begin{eqnarray}
\label{eq:finiteDifferenceDx}\partiald{\X{F}^{n}_{i,j,k}}{x}=\frac{\X{F}^{n}_{i+\hlf,j,k}-\X{F}^{n}_{i-\hlf,j,k}}{\Delta x}\ \ ,\ \ \label{eq:finiteDifferenceDt}\partiald{\X{F}^{n}_{i,j,k}}{t}=\frac{\X{F}^{n+\hlf}_{i,j,k}-\X{F}^{n-\hlf}_{i,j,k}}{\Delta t}
\end{eqnarray}

with relations analogous to \fref{eq:finiteDifferenceDx} for the $y$ and $z$ dimensions. These approximations, applied to the $x$ component of \fref{eq:FDTDH} and rearranged, show

\begin{equation}
\Hat{x}{n+\hlf}{i}{j}{k}=\Hat{x}{n-\hlf}{i}{j}{k} -\mu_0\Delta t \fn{
\frac{\Eat{y}{n}{i}{j}{k+\hlf}-\Eat{y}{n}{i}{j}{k-\hlf}}{\Delta z} - 
\frac{\Eat{z}{n}{i}{j+\hlf}{k}-\Eat{z}{n}{i}{j-\hlf}{k}}{\Delta y}}
\label{eq:finiteDifferenceH}
\end{equation}

A similar analysis may be applied to the other components of $\X{H}$ and $\E$. For \fref{eq:FDTDE}, the time response of the medium (embodied in the dielectric function $\epsilon\fn{t}$), must be explicitly treated. Yee's original FDTD formulation was intended for homogeneous, isotropic and lossless media, but it can equally be applied to materials with complex dielectric functions, and extensions to the method have been developed which can allow limited modelling of non-linear media, as well as the use of so-called conformal meshing techniques. Although the original formulation requires each point in the grid to be defined as a given material, for structures with curving faces for example, conformal meshing allows the dielectric function within a single cell to be composed of a combination of two materials, estimating the contribution of each via the length of the boundary which it occupies. Modifications to the algorithm can necessitate more stringent convergence conditions, and can also cause unintended effects, for example conformal meshing when using metallic media can cause solutions to become divergent, and not converge at all. Another appealing aspect of the FDTD algorithm is that it allows the use not only of constant, or theoretically modelled dielectric functions, such as the Drude-Sommerfeld model explored in \fref{sec:EMmetals}, but also of fits to experimental data, such as those in \cite{JohnsonAndChristy,Palik}.

\begin{figure}%
\begin{center}
      \leavevmode
\includegraphics[width=100mm]{FDTD}%
\caption{An FDTD volume element, or voxel. The cube shows the basic element of the simulation space, known as a Yee cell. The electric field vector elements $\E_{x,y,z}$, are defined at the corners. The elements for this cell are located at $\fn{i,j,k}$, and form three of the cube's edges. The magnetic field vector elements $\X{H}_{x,y,z}$ are defined between adjacent cells' $\E$ field elements, such that each $\E$ element is surrounded by four $\X{H}$ elements and vice versa, in order to facilitate the application of Ampere's and Faraday's laws in the algorithm. In this visualization, the $\X{H}$ vector elements are normals to three of the cube's faces.}%
\label{fig:FdtdVoxel}%
\end{center}
\end{figure}

While the \FDTD\ method allows us to model many varieties of structure with a high degree of accuracy, it inevitably has its limitations. The most immediately apparent to a user of the algorithm are the computational requirements of the method. The \EM\ fields must be computed and temporarily, or in the case of selected points, permanently, stored at every point on the grid. The grid spacing $\Delta x$ must be small enough to resolve both the wavelength of interest, and, more significantly for plasmonic structures which often contain structure much smaller than the free space wavelength, the smallest spatial variations in the structure of interest. Additionally, a condition for convergence of the solution is that the time step $\Delta t$ used in the algorithm must be smaller than the propagation time for the \EM\ field to cross a single cell (this condition is an example of the Courant-Friedrichs-Lewy condition, which is a condition for convergence in numerical solutions of certain differential equations). For the \FDTD\ algorithm for non-dispersive, lossless and isotropic media, can be formulated as 

\begin{equation}
\Delta t \leq \min\left\{\frac{1}{\nu_p\sqrt{\frac{1}{\Delta x^2} + \frac{1}{\Delta y^2} + \frac{1}{\Delta z^2}}}\right\}
\label{eq:CFL}
\end{equation}

Where $\nu_p$ is the wave phase velocity. The condition is more restrictive, and varies by structure and by material, for media which are dispersive, lossy or non-isotropic. It is frequently not useful to determine the exact form of the relation, but simply to reduce the time step found using the condition in \fref{eq:CFL} for the most restrictive point in the mesh by some additional safety factor. The constraints on mesh size and time step size can quickly lead to prohibitively large memory requirements and/or long computation times as structures grow in size and detail, limiting the structures to which the technique can be successfully applied.
 
In the opposite direction, the assumption of continuous media as opposed to discrete atoms or molecules, imposes a constraint on the smallest dimensions which can be successfully modelled. Dimensions approaching the size at which quantum mechanical effects become significant cannot be modelled successfully, as the algorithm uses a purely classical model, and also takes no account of materials' substructure.

%\subsection{Other Numerical Techniques}
%\label{sec:OtherNumerical}
%\Check{I don't know much about them - is this section worth including, or should I skip it to save space?}

\section{Sample Fabrication}
\label{sec:Fabrication}
The most common methods for creating plasmonic structures are lithographic processes, either photolithography, or \EBL. However, in certain cases other methods such as \FIB\ milling can be more appropriate.

\subsection{Electron beam lithography}
\label{sec:EBL}
As the name suggests, electron beam lithography is essentially a printing process, whereby a pattern is created in a resist using a beam of electrons, and then transferred to another material. The essentials of the technique are outlined in \fref{fig:EBL}.

\begin{figure}%
\begin{center}
      \leavevmode
\includegraphics[width=\textwidth]{EBL}%
\caption{Electron beam lithography process. A thin (\nm{100} or so at the centre) layer of resist (we use ZEP520A) is added to the substrate using a spin-coater. The resist is exposed using an electron beam (Crestec 9500C), which breaks polymer bonds. During development, the exposed resist is removed in hexyl acetate. The result is a layer of resist with holes in, forming a mask over the substrate. A thin adhesion layer of Cr (\nm{1-5}), followed by the desired thickness of Au is evaporated onto the sample. through the mask. Finally, the remaining resist is removed using Shipley Remover 1165. The gold film on the resist surface floats off into the remover solution, leaving the desired gold structures on the substrate.}%
\label{fig:EBL}%
\end{center}
\end{figure}

The use of the electron beam means that the diffraction-limited beam size is much smaller than for light, and as a result allows much smaller features to be created. beam diameters of around \nm{5} are not uncommon, however scattering of secondary electrons in the resist and substrate can limit the effective feature size obtainable to a slightly higher value of around \nm{10} or so.

In order for an electron-beam exposure to be performed successfully, the sample must conduct. If the sample does not conduct, charge build-up from the electron beam on the sample surface leads to the beam being deflected from its intended path, exposing the wrong areas of the sample, and ruining the lithography pattern. The simplest way of solving this problem is to use a conducting substrate, such as Silicon (which is conductive enough at the high voltages used for the electron beam) or \ITO. Alternatively, a non-conducting substrate can be coated with a thin conducting layer, as with \ITO-coated glass. However, for plasmonic structures, the conductivity $\sigma$ of the substrate, which is related in the frequency domain to the material's dielectric function by $\epsilon\fn{\w}=1+i\sigma\fn{w}/\fn{\epsilon_0\w}$, can cause significant modification to the structure's plasmonic response, and as a result we are often in a situation in which we need to use a fully insulating substrate such as quartz. In such cases, we deposit a thin (\nm{\sim\!10}) layer of Al over the resist before exposure. 
 
\subsection{Photolithography}
\label{sec:photolith}
Photolithography functions in a similar manner to \EBL\ outlined above, except it uses UV light, rather than an electron beam, to expose the resist. The light is applied through a metal mask, usually created using \EBL\ and chemical etching of a chromium-coated glass slide. The holes in the metal mask define which areas of the resist are exposed to the UV lamp. A single mask can contain around a dozen or so different patterns at different locations. Since it uses UV light for the exposure, the minimum feature sizes which are possible using photolithography are limited by diffraction to a few hundred \nm{}\footnote{although in the semiconductor industry, the use of near field optics allows corrections to be made such that patterns with feature sizes significantly smaller than the diffraction limit can be created, this is still not sufficient for the plasmonic structures we are producing.}. In addition, a photolithography process requires a mask to be made using \EBL. Once produced, the mask is fixed, and so photolithography is not suited to applications involving frequently varying structures, or patterns which will only be exposed once or twice. However, photolithography does not require a conducting sample, and has very quick exposure times (and entire pattern can be exposed in seconds, even for very large feature sizes). A photolithography mask aligner is also considerably cheaper to use than an electron column, once the initial cost of the mask is covered. As a result, for larger structures which are required frequently, photolithography is a more favourable option than \EBL. We make use of photolithography for applications such as creating \IDTs\ (structures for generating surface acoustic waves) and their associated wirebond pads, large waveguide structures\footnote{Here I denote a structure as large when its smallest feature is larger than \um{0.5}, approximately the minimum feature size which can be resolved with our photolithography process.}, and simple grids of markers which, we use to relocate areas of interest on a sample. 














\subsection{Chemical synthesis of monocrystalline Au}
\label{sec:Alchemy}
The production of metallic nanoparticles, particularly for noble metals, has been a research interest of surface chemists for some time. Metal nanoparticles are of significant scientific and technological interest due to their unique optical responses, as demonstrated in surface enhanced Raman scattering studies, their catalytic action and unusual chemical properties. Much study has been devoted to achieving smaller distributions of particle size, and simpler synthesis techniques. A variety of morphologies have been produced, including spheres, sheets, rods and wires, with most protocols involving the reduction of metallic ions in either aqueous or organic solvents\cite{Chu2006,Tsuji2005}.

We are most interested, of course, in noble metal particles, particularly in gold. Of most interest to us as a construction material for plasmonic structures are a class of particles referred to as gold nanoplates. These structures are sheets of monocrystalline gold, with thicknesses of around \nm{10-100}, and diameters between a few dozen \nm{\!} and \um{40}, depending on growth conditions\cite{Chu2006,Tsuji2005}. The plates can be triangular, hexagonal, or triangular with truncated corners, usually a mix of all three. From selected area electron diffraction (SAED) measurements, the plates appear monocrystalline, with the large flat faces on the \{1,1,1\} crystal lattice vector\cite{Sun2005,Guo2006,Mallikarjuna2007,Chu2006}.

Initial synthesis methods involved reduction of chloroauric acid ($\rm HAuCl_4$) in ethylene glycol. Poly-vinyl pyrrolidone (PVP) is added, functioning both as a reducing agent and as a capping agent\cite{Xiong2006}. PVP adsorbed preferentially onto the \{1,1,1\} face, preventing growth along this axis and leading to the anisotropic plates. Subsequent studies have used other reagents, including aqueous solutions of sugars\cite{Mallikarjuna2007}, as well as aniline or polyamines in place of PVP\cite{Guo2006,Sun2005}.

The PVP process is carried out in an oil bath at temperatures between $65$ and \unit{200}{^\circ C}. Studies on the effects of microwave heating suggest that the occurrence of burst nucleation due to rapid, homogeneous heating of the solvent, combined with hot-surface effects caused by adsorption of PVP onto the plate surface, may promote plate growth compared to oil bath heating \cite{Tsuji2005}. Simpler synthesis routes include the aniline and polyamine processes, both of which can be performed satisfactorily at \unit{100}{^\circ C} or below, and thus permit the use of a water bath rather than an oil bath. Plates produced using these processes can routinely reach diameters of \um{15}, with sizes of up to \um{40} reported in \cite{Sun2005}.

For our purposes, a simple protocol for fabricating large diameter plates is desirable, and as a result we began with the aniline-based method of \cite{Guo2006}. Flake synthesis is a simple process, and proceeds typically as follows. \unit{50}{ml} Ethylene glycol containing chloroauric acid is heated to \unit{95}{^\circ C} for 20 minutes using a water bath. 0.1M aniline is added to the solution to obtain a 2:1 molar ratio to gold,with moderate stirring. The solution colour changes quickly upon adding the aniline, but it is kept heating typically for several hours, after which the flakes are washed and suspended in anhydrous ethanol for storage. Deposition onto a substrate is accomplished by re-suspending the flakes by ultrasonication, then allowing a drop of the ethanol to evaporate from the substrate in air, leaving the flakes attached to the substrate surface. Microwave heating can produce flakes in as little as 30-60 seconds, but their small size makes them less useful to us at precursors for waveguide structures. 

The flake growth process also produces a large quantity of nanometer scale spherical gold particles, which are problematic. As a result, we pass the suspended flakes through a \um{5} syringe filter before use, keeping only the large fraction. Flakes synthesized by Dr Yury Alaverdyan using the aniline process can be seen in \fref{fig:flakes}. 


















\subsection{Focused Ion Beam}
\label{sec:FIB}
In a similar manner to the electron beam used for \EBL, a focussed ion beam, or FIB, is an instrument which produces a beam of positive ions, which are focussed and directed by a series of electrostatic lenses. The most common type of ion source in commercially produced instruments, including the Zeiss instrument in the Institute for Manufacturing (IfM) which we have used, is a liquid metal source, often using Gallium. Gallium is heated on a tungsten needle, the liquid metal Gallium forming a sharp conical tip\footnote{Elemental Gallium melts at around \unit{30}{^\circ C}}. A very large electric field is applied to the tip, causing ionization and field emission of ${\rm Ga}^+$ ions.

Used at low currents (a few ${\rm pA}$), the ion beam can be used to image a sample in much the same way as a scanning electron microscope (SEM). The impact of the Ga ions generates secondary electrons, which can be detected in addition to ions, to form an image as the beam is raster-scanned across the sample. Given the much higher momentum of the ions compared to an equivalent electron beam, \FIB\ imaging is inherently a destructive process, however at low current the physical damage to the sample can be minimal. At higher currents (several to tens of ${\rm nA}$), the ion beam physically sputters material from the sample, and can be used for micro-machining, a common use being to use the ion beam to cut thin membranes from a sample for TEM analysis. Some instruments, including that available to us at the IfM, incorporate both an ion beam for milling structures and an SEM to non-destructively image the milled structures. \FIB\ systems can also incorporate gas-assisted deposition systems, whereby the ion beam is used to break down a volatile gas at selected areas of the sample, leading to deposition of one of the gas' components. This is used with gold and platinum, for example, in the semiconductor industry, to allow post-production modification of circuit prototypes. However, the purity of deposited material is low, due to incorporation of both Ga and quantities of the volatile organic compounds used to provide the deposition material. As such, gas deposition systems do not present a viable route for the creation of plasmonic structures.

Milling plasmonic structures such as waveguides and antennae from a pre-prepared metal film is possible though, and indeed may provide advantages over existing EBL production methods. The sputtering rate of a \FIB\ is dependent on the direction along the crystal lattice in which the material is being milled. For thermally-deposited films therefore, different grains, which are present at random orientations, are milled at different rates, leading to a rough surface after milling, even when the starting surface is considerably smoother. Milling of monocrystalline films obviously does not suffer from this problem, as the crystal lattice is uniform across the film. However, redeposition of sputtered material, as well as incorporation of Gallium into the film, will occur at the edges of milled structures. By using a \FIB\ to mill structures from chemically synthesized monocrystalline flakes, it may be possible to create structures which operate with minimum losses from roughness and grain boundaries.

\section{Optical measurements}
\label{sec:optics}
\subsection{Confocal microscopy}
\label{sec:confocal}
The principle of confocal microscopy relies upon limiting the volume  both of the excitation and collection. A small excitation area is achieved by using a gaussian beam to illuminate the objective. This produces a diffraction-limited focal spot of diameter $\lambda/\fn{\rm2\ NA}$ in the objective focal plane. The gaussian beam for our microscope is produced by collimating the output of a single-mode optical fibre, which has a core diameter of \Check{\um{28}}, forming a pinhole. The excitation volume is limited to a similar region in an analogous manner, by limiting collected light using a pinhole in the focal plane of the tube lens. In our microscope, the `pinhole' is the input of another single-mode optical fibre. This spatial filter removes light from regions of the sample which are outside the focal spot in the sample plane, and greatly attenuates light from points within the focal spot but out of focus. Images are formed by raster scanning the sample relative to the focal spot (in our microscope, it is the spot which is scanned across a stationary sample) and recording intensities at each point.

In general, as with our PL measurements, it is desired to collect light from the region which is excited, and thus the excitation and collection arms must be aligned to overlap the focal spots. However, for our measurements of the transmission of waveguides, we wanted to excite one end of the waveguide, while collecting light from the other end. As a result, we deliberately `misaligned' the microscope in order to displace the two focal spots relative to each other. Scanning the mirror SM in this setup actually scans both focal spots simultaneously across the sample, maintaining their separation. With the separation set to the waveguide length, the result is a scan which shows a bright line at the point where excitation hits one end of the waveguide and collection is from the other.

For our microscope, a schematic of which is presented in \fref{fig:confocal}, excitation is introduced through a single-mode fibre of appropriate wavelength range, depending on the application. Other optics are also altered. For PL studies, excitation is by a \nm{532} CW laser (Verdi, Coherent), with filter F1 (a \nm{532} laser line filter) used to remove longer-wavelength components generated by scattering processes in the fibre. LP and $\lambda/2$ denote linear polarizers and half wave-plates respectively, allowing adjustment of excitation and collection polarizations. All are for NIR use. For PL measurements, which use green excitation, polarization of excitation is not set, since PL from gold films is unpolarized, regardless of excitation polarization. Filter F2 is a \nm{600} long-pass filter, which removes the excitation light for PL measurements. BS denotes two beam splitters, upper (in this figure) is uncoated glass, the other is a beam sampler with $4^\circ$ wedge angle and an AR coating on the reverse side. The excitation arm is passed into the photodiode PD in order to maintain constant excitation power between measurements. The collection arm is also fibre-coupled, the input of the single-mode fibre forming the spatial filter of the confocal setup. Collected intensity is monitored using an avalanche photodiode (APD) single photon counter. SM denotes a scanning mirror, positioned by piezo actuators, and controlled by a computer. Altering the angle of this mirror scans the location of the focal spot across the sample, with the full scan range covering \um{\sim 100}. For the waveguide transmission studies, which required longer wavelengths, a \nm{780} diode laser was used for excitation in place of the Verdi. The filter F1 was consequently removed for these measurements. The microscope objective has an NA of 0.9, and is a long working distance NIR-corrected lens.

\begin{figure}%
\begin{center}
\leavevmode
\includegraphics[width=100mm]{confocal}%
\caption{A schematic of our confocal microscopy setup. The same microscope is used for both PL measurements and waveguide transmission measurements. Different filters and polarization optics are required for each measurement type, as described in the text.}%
\label{fig:confocal}%
\end{center}
\end{figure}
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